Appendix: Reminders

RemindersHere, we provide the precise statements of some definitions and results used from Volume 1, Banach, Fréchet, Hilbert and Neumann Spaces [72], Volume 2, Continuous Functions [73], and Volume 3, Distributions [74] of the Analysis for PDEs set, which are used in this book in order to make it self-contained 1 .The proofs are given in the aforementioned volumes, to which we refer, for example by [Vol. 1, Theorem 12.2].Some statements are restricted to the cases relevant to us when this simplifies them, whereas others are completed and then subsequently proven. A.1. Notation and numbering Set theory notationWe use the standard set theory notation, namely, = for "is equal to", ∈ for "belongs to", ∀ for "for all", ∃ for "there exists", ⇒ for "implies", ⇐ for "follows from", ⇔ for "is equivalent to".A barred symbol denotes the negation of a property; for example, ̸ = means "is not equal to".We denote by {a, b, . . ., z} the set consisting of a, b, . . ., z, by {a : P (a)} or {a} P (a) the set of all a satisfying the property P (a), and by ∅ the empty set.We write ⊂ for set inclusion, ∪ for union, and ∩ for intersection.We write ⊃ for "contains", andWe denote by (a, b, . . ., z) the ordered set consisting of a, b, . . ., z.Thus, (a, b) ̸ = (b, a) if a ̸ = b.In contrast, we have {a, b} = {b, a}. We denote byGiven a set I and, for each i ∈ I, a set U i , we denote by i∈I U i the union of the U i , by i∈I U i their intersection, and by i∈I U i their product, namely, the set {(u i ) i∈I : u i ∈ U i , ∀i}.A mapping from a set U into a set X is the assignment, to each u ∈ U , of an element T (u) ∈ X.We say that T (u) is the image under T of the point u.The image of a subset V of U under T is the set1 Historical note.In Volumes 1, 2, and 3, the reader will find the historical background of many of the definitions and properties recalled here.